Glasgow all-but secured their Celtic Cup spot with a remarkable, seven-try win at a stunned Stradey Park.
Scarlets (18) 30
Tries: Havili (2), Selley, Wyatt, Davies
Glasgow (28) 57
Tries: Swindall, R Lamont (3), S Lamont (2), Pinder
Cons: Parks (5)
Pens: Parks (4)
Rory Lamont claimed a hat trick, stealing the headlines from brother Sean who crossed for two tries.
Steve Swindall and Sam Pinder got the other tries, with fly-half Sam Parks having an outstanding day with the boot to add a further 22 points.
The Scarlets gained a bonus point with tries from Aisea Havili (2), Tal Selley, Chris Wyatt and Barry Davies.
That means the Llanelli side have still not secured their Celtic Cup spot and their chances of a home draw in that competition are slim.
It was the reigning champions' heaviest ever defeat in the Celtic League and the first time that Glasgow had passed 50 points in the competition.
Playing against a strong wind, the Scarlets made a nightmare start, conceding two early penalties to Parks.
His tactical kicking was also working, pegging the home team into their 22.
From the good field position, blind-side Swindall powered over with a short-range drive.
The Scarlets hit back with a Gareth Bowen penalty, then fine handling from Matthew Watkins and Wyatt sent Havili over for a try in the corner.
Parks added another penalty, before a break from a scrum in the Scarlets half saw Dwayne Peel free Selley for a 40m sprint to the line for the Welsh side.
That seemed to give the Scarlets the platform to take the game, but Glasgow won quick maul ball and spread it to the wing where Rory Lamont had the strength and pace to cross.
The home team fired back and a Bowen break gave them position on the Scottish line, but Peel's scoring pass was intercepted by the other Lamont, Sean, who sprinted the length of the field for a try to make it 28-13.
The Scarlets desperately needed a score, and after a series of forward charges Wyatt bullied his way over for a try on the stroke of half time.
But it was all Glasgow after the break, Rory Lamont crossing after great inter-play between forwards and backs.
Davies dropped a try-scoring chance over the line, before turnover ball fell to Sean Lamont who powered home from 60m.
The Scarlets chose to kick a penalty to touch near the Glasgow try line, but to the disbelief of the home crowd Bowen chipped it straight to Pinder.
The scrum-half's gallop for the line from 90m was hauled back, but the ball was recycled and came back to him to cross under the posts.
Gavin Thomas' break sent Davies in for the consolation of a bonus-point try for the home team and Havili added his second score.
But the final, hat-trick try went to Rory Lamont after another devastating attacking burst by Glasgow.
"It was just one of those games," said the hat-trick man. "There were so many breakaway tries that could have gone either way.
"We managed to get the scores at crucial times when we were under pressure."
His brother Sean added: "We like the open style of rugby and for once we were able to pull it off away from home."
Scarlets: B. Davies, Havili, Watkins, Finau, Selley, Bowen, Peel, John, Rees, J. Davies, Cooper, Wyatt, Easterby, G. Thomas, Powell.
Replacements: Phillips for Havili (80), G. Evans for Watkins (67), S. Thomas for Finau (77), McBryde for Rees (63), A. Jones for Wyatt (63). Not Used: Maddocks, G. Quinnell.
Glasgow: Howarth, R. Lamont, Craig, Henderson, S. Lamont, Parks, Pinder, Tckachuk, Lawson, Murray, Beardshaw, Turner, Swindall, Petrie, Wilson.
Replacements: Shaw for Howarth (59), Beveridge for Pinder (66), Hall for Beardshaw (69), Dearlove for Swindall (56). Not Used: Bulloch, Prescott, Barrow.
Referee: Simon McDowell (Ireland).